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Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
Weyl’s type theorems and
perturbations
Teoremas de tipo Weyl y perturbaciones
Pietro Aiena ([email protected])
Dipartimento di Matematica ed Applicazioni
Facoltà di Ingegneria, Università di Palermo
Viale delle Scienze, I-90128 Palermo (Italy)
Jesús R. Guillen ([email protected])
Departamento de Matemáticas, Facultad de Ciencias
ULA; Merida (Venezuela)
Pedro Peña ([email protected])
Departamento de Matemáticas, Facultad de Ciencias
ULA; Merida (Venezuela)
Abstract
Weyl’s theorem for a bounded linear operator T on complex Banach
spaces, as well as its variants, a-Weyl’s theorem and property (w), in
general is not transmitted to the perturbation T + K, even when K
is a ”good” operator, as a commuting finite rank operator or compact
operator. Weyl’s theorems do not survive also if K is a commuting
quasi-nilpotent operator. In this paper we discuss some sufficient conditions for which Weyl’s theorem, a-Weyl’s theorem as well as property
(w) is transmitted under such kinds of perturbations.
Key words and phrases: Local spectral theory, Fredholm theory,
Weyl’s theorem.
Resumen
El Teorema de Weyl para un operador lineal acotado T sobre un
espacio de Banach complejo, como también sus variantes, el teorema
a-Weyl y la propiedad (w), en general no se transmite a la perturbación
T + K, ni aún cuando K es un ”buen” operador, como operadores
Received 2006/03/12. Revised 2006/07/02. Accepted 2006/07/22.
MSC Primary 47A10, 47A11. Secondary 47A53, 47A55.
56
Pietro Aiena, Jesús R. Guillen, Pedro Peña
conmutantes de rango finito u operadores compactos. Los teoremas de
Weyl no sobreviven tampoco si K es un operador conmutante cuasinilpotente. En este artı́culo discutimos algunas condiciones suficientes
para las cuales el teorema de Weyl, el teorema a-Weyl y la propiedad
(w) se transmite bajo tales tipos de perturbaciones.
Palabras y frases clave: Teorı́a espectral local, teorı́a de Fredholm,
teorema de Weyl.
1
Introduction and preliminaries
Weyl [35] examined the spectra of all compact perturbations of a hermitian
operator T on a Hilbert space and proved that their intersection coincides
with the isolated point of the spectrum σ(T ) which are eigenvalues of finite
multiplicity. Weyl’s theorem has been extended to several classes of Hilbert
spaces operators and Banach spaces operators. More recently, two variants
of Weyl’s theorem have been introduced by Rakočević [32], [31], the so called
a-Weyl’s theorem and the property (w) studied also in [10].
In general, Weyl’s theorem is not sufficient for Weyl’s theorem for T + K,
where K ∈ L(X). Weyl’s theorems are liable to fail also under ”small” perturbations if ”small” is interpreted in the sense of compact or quasi-nilpotent
operators. In this note, we study sufficient conditions for which we have the
stability of Weyl’s theorems and property (w), under perturbations by finite
rank operators, compact operators, or quasi-nilpotent operator commuting
with T . We shall see that the stability of Weyl’s theorems requires some
special conditions on the isolated points of the spectrum (or on the isolated
points of the approximate point spectrum).
We begin with some standard notations on Fredholm theory. Throughout
this note, let L(X) denote the algebra of all bounded linear operators acting
on an infinite dimensional complex Banach space X. If T ∈ L(X) write α(T )
for the dimension of the kernel ker T and β(T ) for the codimension of the
range T (X). Denote by
Φ+ (X) := {T ∈ L(X) : α(T ) < ∞ and T (X) is closed}
the class of all upper semi-Fredholm operators, and by
Φ− (X) := {T ∈ L(X) : β(T ) < ∞}
the class of all lower semi-Fredholm operators. The class of all semi-Fredholm
operators is defined by Φ± (X) := Φ+ (X) ∪ Φ− (X), while the class of all
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Weyl’s type theorems and perturbations
57
Fredholm operators is defined by Φ(X) := Φ+ (X) ∩ Φ− (X). The index of
a semi-Fredholm operator is defined by ind T := α(T ) − β(T ). Recall that
the ascent p := p(T ) of a linear operator T is defined to be the smallest nonnegative integer p such that ker T p = ker T p+1 (X). If such an integer does
not exist we put p(T ) = ∞. Analogously, the descent q := q(T ) of an operator
T is the smallest non-negative integer q such that T q (X) = T q+1 (X), and if
such an integer does not exist we put q(T ) = ∞. It is well-known that if p(T )
and q(T ) are both finite then p(T ) = q(T ), [1, Theorem 3.3]. Two important
classes of operators are the class of all upper semi-Browder operators
B+ (X) := {T ∈ Φ+ (X) : p(T ) < ∞},
and the class of all lower semi-Browder operators
B− (X) := {T ∈ Φ− (X) : q(T ) < ∞}.
The class of all Browder operators is defined by B(X) := B+ (X) ∩ B− (X).
Recall that a bounded operator T ∈ L(X) is said to be a Weyl operator if
T ∈ Φ(X) and ind T = 0. Clearly, if T is Browder then T is Weyl, since the
finiteness of p(T ) and q(T ) implies, for a Fredholm operator, that T has index
0, see [1, Theorem 3.4].
These classes of operators motivate the definition of several spectra. The
upper semi-Browder spectrum of T ∈ L(X) is defined by
σub (T ) := {λ ∈ C : λI − T ∈
/ B+ (X)},
the lower semi-Browder spectrum of T ∈ L(X) is defined by
σlb (T ) := {λ ∈ C : λI − T ∈
/ B− (X)},
while the Browder spectrum of T ∈ L(X) is defined by
σb (T ) := {λ ∈ C : λI − T ∈
/ B(X)}.
The Weyl spectrum of T ∈ L(X) is defined by
σw (T ) := {λ ∈ C : λI − T is not Weyl}.
We have that σw (T ) = σw (T ∗ ), where T ∗ denotes the dual of T , while
σub (T ) = σlb (T ∗ ),
σlb (T ) = σub (T ∗ ).
Evidently,
σw (T ) ⊆ σb (T ) = σw (T ) ∪ acc σ(T ),
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
where we write acc K for the accumulation points of K ⊆ C, see [1, Chapter
3]. The Weyl (or essential) approximate point spectrum σwa (T ) of a bounded
operator T ∈ L(X) is the complement of those λ ∈ C for which λI − T ∈
Φ+ (X) and ind (λI − T ) ≤ 0. Note that σwa (T ) is the intersection of all
approximate point spectra σa (T + K) of compact perturbations K of T , see
[32]. The Weyl surjectivity spectrum σws (T ) is the complement of those λ ∈ C
for which λI − T ∈ Φ− (X) and ind (λI − T ) ≥ 0. The spectrum σwa (T )
coincides with the intersection of all surjectivity spectra σs (T +K) of compact
perturbations K of T , see [32] or [1, p.151]. Clearly, the last two spectra are
dual each other, i.e.,
σwa (T ) = σws (T ⋆ ) and
σws (T ) = σwa (T ⋆ ).
Furthermore, σw (T ) = σwa (T )∪σws (T ). Since p(T ) < ∞ (respectively, q(T ) <
∞) entails that ind T ≤ 0 (respectively, ind T ≥ 0), we have σwa (T ) ⊆ σub (T )
and σws (T ) ⊆ σlb (T ), and the precise relationship between these spectra is
given by the following equalities:
σub (T ) = σwa (T ) ∪ acc σa (T ),
(1)
σlb (T ) = σws (T ) ∪ acc σs (T ),
(2)
and
see [33]. This article also deals with the single valued extension property. This
property has a basic role in the local spectral theory, see the recent monograph
of Laursen and Neumann [24] or Aiena [1]. In this article we shall consider a
localized version of this property, recently studied by several authors [8], [5],
[9], and previously by Finch [20], and Mbekhta [26].
Definition 1.1. Let X be a complex Banach space and T ∈ L(X). The
operator T is said to have the single valued extension property at λ0 ∈ C
(abbreviated SVEP at λ0 ), if for every open disc U of λ0 , and f : U → X
satisfying the equation (λI − T )f (λ) = 0 for all λ ∈ U we have f ≡ 0 on U .
An operator T ∈ L(X) is said to have the SVEP if T has the SVEP at every
point λ ∈ C.
An operator T ∈ L(X) has the SVEP at every point of the resolvent
ρ(T ) := C \ σ(T ). The identity theorem for analytic function ensures that
for every T ∈ L(X), both T and T ∗ have the SVEP at the points of the
boundary ∂σ(T ) of the spectrum σ(T ). In particular, that both T and T ∗
have SVEP at every isolated point of σ(T ) = σ(T ∗ ). The SVEP is inherited
by the restrictions to closed invariant subspaces, i.e. if T ∈ L(X) has the
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Weyl’s type theorems and perturbations
59
SVEP at λ0 and M is a closed T -invariant subspace then T |M has SVEP at
λ0 .
Recall that T ∈ L(X) is said to be bounded below if T is injective and has
closed range. Let σa (T ) denote the classical approximate point spectrum of T ,
i. e. the set
σa (T ) := {λ ∈ C : λI − T is not bounded below},
and let
σs (T ) := {λ ∈ C : λI − T is not surjective}
denote the surjectivity spectrum of T . Note that
p(λI − T ) < ∞ ⇒ T has SVEP at λ,
(3)
q(λI − T ) < ∞ ⇒ T ∗ has SVEP at λ,
(4)
and dually
see [1, Theorem 3.8]. From the definition of SVEP we also have
σa (T ) does not cluster at λ ⇒ T has SVEP at λ,
(5)
σs (T ) does not cluster at λ ⇒ T ∗ has SVEP at λ.
(6)
and dually
The quasi-nilpotent part of T is defined by
1
H0 (T ) := {x ∈ X : lim kT n xk n = 0}.
n→∞
It is easily seen that H0 (T ) is a linear T -invariant subspace and ker (T m ) ⊆
H0 (T ) for every m ∈ N. The subspace H0 (T ) may be not closed and coincides
with the local spectral subspace XT ({0} in the case where T has SVEP, see
Theorem 2.20 of [1]. The following implication holds for every operator T ∈
L(X),
H0 (λI − T ) is closed ⇒ T has SVEP at λ,
(7)
see [5].
Remark 1.2. There is a fundamental relationship between SVEP and the Fredholm theory: all the implications (3)-(7) become equivalences if we assume
that λI − T ∈ Φ± (X), see [5], [9].
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2
Pietro Aiena, Jesús R. Guillen, Pedro Peña
Weyl’s theorem under perturbations
If T ∈ L(X) let us consider the complement in σ(T ) of the Browder spectrum:
p00 (T ) := σ(T ) \ σb (T ) = {λ ∈ σ(T ) : λI − T is Browder}.
Write iso K for the set of all isolated points of K ⊆ C, and set
π00 (T ) := {λ ∈ iso σ(T ) : 0 < α(λI − T ) < ∞},
the set of isolated eigenvalues of finite multiplicities. Obviously,
p00 (T ) ⊆ π00 (T ) for every T ∈ L(X).
(8)
Following Coburn [15], we say that Weyl’s theorem holds for T ∈ L(X) if
∆(T ) := σ(T ) \ σw (T ) = π00 (T ),
(9)
while T is said to satisfy Browder’s theorem if
σ(T ) \ σw (T ) = p00 (T ),
or equivalently, σw (T ) = σb (T ). Note that
Weyl’s theorem ⇒ Browder’s theorem,
see, for instance [1, p. 166]. Browder’s theorem may be characterized by
means of SVEP:
T satisfies Browder’s theorem ⇔ T has SVEP at all λ ∈
/ σw (T ).
Browder’s theorem corresponds to the half of Weyl’s theorem, in the following sense:
Theorem 2.1. [2] If T ∈ L(X) then the following assertions are equivalent:
(i) Weyl’s theorem holds for T ;
(ii) T satisfies Browder’s theorem and π00 (T ) = p00 (T );
The conditions p00 (T ) = π00 (T ) is equivalent to several other conditions,
see [1, Theorem 3.84]. The equality p00 (T ) = π00 (T ) may be described in
another way. To see this, let P0 (X), where X is a Banach space, denote the
class of all operators T ∈ L(X) such that there exists p := p(λ) ∈ N for which
H0 (λI − T ) = ker (λI − T )p
for all λ ∈ π00 (T ).
Then we have
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
(10)
Weyl’s type theorems and perturbations
61
Theorem 2.2. [6] With the notation above, T ∈ P0 (X) if and only if p00 (T ) =
π00 (T ). In particular, if T has SVEP then Weyl’s theorem holds for T if and
only if T ∈ P0 (X).
The following classes of operators has been introduced by Oudghiri [28],
see also [11].
Definition 2.3. A bounded operator T ∈ L(X) is said to satisfy property
H(p) if
H0 (λI − T ) = ker (λI − T )p for all λ ∈ C.
(11)
for some p = p(λ) ∈ N, .
From the implication in (7) it is clear that property H(p) entails SVEP.
Many of the commonly considered operators on Banach spaces and Hilbert
spaces have property H(p), or have SVEP and belong to the class P0 (X). We
list now some classes of operators for which property H(p) holds. In the sequel
by X we shall denote a Banach space, while a Hilbert space will be denoted
by H.
(A) A bounded operator T ∈ L(X) is said to be paranormal if
kT xk2 ≤ kT 2 xkkxk for all x ∈ X.
The class of paranormal operators properly contains a relevant number of
Hilbert space operators, among them p-hyponormal operators, log-hyponormal
operators, quasi-hyponormal operators (see later for definitions). Every paranormal operator T ∈ L(H) has SVEP. This may be easily seen as follows: if
λ 6= 0 and λ 6= µ then, by Theorem 2.6 of [14], we have kx + yk ≥ kyk whenever x ∈ ker (µI − T ) and y ∈ ker (λI − T ). It then follows that if U is an open
disc and f : U → X is an analyitic function such that 0 6= f (z) ∈ ker (zI − T )
for all z ∈ U , then f fails to be continuous at every 0 6= λ ∈ U .
(B) An operator T ∈ L(X) is called totally paranormal if λI − T is paranormal for all λ ∈ C. Every totally paranormal operator satisfies property
H(1), see [11]. In the sequel we shall denote by T ′ the Hilbert adjoint of
T ∈ L(H).
An operator T ∈ L(H) is said to be *-paranormal if
kT ′ xk2 ≤ kT 2 xk
holds for all unit vectors x ∈ H. T ∈ L(H) is said to be totally *-paranormal
if λI − T is *-paranormal for all λ ∈ C. Every totally ∗-paranormal operator
satisfies property H(1), see [23]. The class of totally paranormal operators includes all hyponormal operators on Hilbert spaces H. Recall that the operator
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
T ∈ L(H) is said to be hyponormal if
kT ′ xk ≤ kT xk
for all x ∈ X.
Hyponormal operators are p-hyponormal with p = 1, where T ∈ L(H) is said
to be p-hyponormal, with 0 < p ≤ 1, if
(T ′ T )p ≥ (T T ′ )p .
Every injective p-hyponormal operator satisfies property H(1), see [11]. The
concept of hyponormality may be relaxed as follows: T ∈ L(H) is said to be
quasi-hyponormal if
2
(T ′ T )2 ≤ T ′ T 2
Also quasi-normal operators are totally paranormal, since these operators are
hyponormal, see Conway [16] for details.
(C) An operator T ∈ L(H) is said to be log-hyponormal if T is invertible
and satisfies
log (T ′ T ) ≥ log (T T ′ ).
Every log-hyponormal operator satisfies the condition H(1), see [11]. In fact,
log-hyponormal operator are similar to hyponormal operators and property
H(1) is preserved by similarity [11, Theorem 2.4].
(D) Let A be a commutative Banach algebra. A linear map T : A → A
is said to be a multiplier if (T x)y = x(T y) holds for all x, y ∈ A. If A is a
commutative semi-simple Banach algebra, then every multiplier satisfies the
condition H(1) see [5]. Moreover, every multiplier of a commutative semisimple Banach algebra has SVEP, [1, Chapter 4]. In particular, the condition
H(1) holds for every convolution operator on the group algebra L1 (G), where
G is a locally compact abelian group.
(E) An important class of operators which satisfy property H(p) is given
by the class of subscalar operators [28]. Recall that T ∈ L(X) is said to
be generalized scalar if there exists a continuous algebra homomorphism Ψ :
C ∞ (C) → L(X) such that
Ψ(1) = I
and Ψ(Z) = T,
where C ∞ (C) denote the Fréchet algebra of all infinitely differentiable complexvalued functions on C, and Z denotes the identity function on C. Every generalized scalar operator has SVEP, see [24]. An operator is subscalar whenever
is similar to the restriction of a generalized scalar operator to one of its closed
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Weyl’s type theorems and perturbations
63
invariant subspaces. The interested reader can find a well organized study of
these operators in the book by Laursen and Neumann [24]. Property H(p)
is then satisfied by p-hyponormal operators and log-hyponormal operators,
M -hyponormal operators [24, Proposition 2.4.9], and totally paranormal operators [11], since all these operators are subscalar.
(F) An operator T ∈ L(X) for which there exists a complex non-constant
polynomial h such that h(T ) is paranormal is said to be algebraically paranormal. If T ∈ L(H) is algebraically paranormal then T ∈ P0 (H), so it satisfies
the condition (10), see [2], but in general the condition H(p) is not satisfied
by paranormal operators, (for an example see [6, Example 2.3]). Since every paranormal operator has SVEP then also every algebraically paranormal
operator has SVEP, see Theorem 2.40 of [1] or [18], so that Weyl’s theorem
holds for every algebraically paranormal operator, see also [21].
Weyl’s theorem is transmitted by some special perturbations. The following result has been first proved by Oberai [27].
Theorem 2.4. Weyl’s theorem is transmitted from T ∈ L(X) to T + N when
N is a nilpotent operator commuting with T .
The following example shows that Oberai’s result does not hold if we do
not assume that the nilpotent operator N commutes with T
Example 2.5. Let X := ℓ2 (N) and T and N be defined by
T (x1 , x2 , . . . , ) := (0,
x1 x2
, , . . . ),
2 3
(xn ) ∈ ℓ2 (N)
and
x1
, 0, 0, . . . ), (xn ) ∈ ℓ2 (N)
2
Clearly, N is a nilpotent operator, and T is a quasi-nilpotent operator satisfying Weyl’s theorem. On the other hand, it is easily seen that 0 ∈ π00 (T + N )
and 0 ∈
/ σ(T +N )\σw (T +N ), so that T +N does not satisfies Weyl’s theorem.
N (x1 , x2 , . . . , ) := (0, −
Note that the operator N in Example 2.5 is also a finite rank operator not
commuting with T . In general, Weyl’s theorem is also not transmitted under
commuting finite rank perturbation.
Example 2.6. Let S : ℓ2 (N) → ℓ2 (N) be an injective quasi-nilpotent operator, and let U : ℓ2 (N) → ℓ2 (N) be defined :
U (x1 , x2 , . . . , ) := (−x1 , 0, 0, . . . ),
with (xn ) ∈ ℓ2 (N).
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
Define on X := ℓ2 (N) ⊕ ℓ2 (N) the operators T and K by
T := I ⊕ S
and K := U ⊕ 0
Clearly, K is a finite rank operator and KT = T K. It is easy to check that
σ(T ) = σw (T ) = σa (T ) = {0, 1}.
Now, both T and T ∗ have SVEP, since σ(T ) = σ(T ∗ ) is finite. Moreover,
π00 (T ) = σ(T ) \ σw (T ) = ∅, so T satisfies Weyl’s theorem.
On the other hand,
σ(T + K) = σw (T + K) = {0, 1},
and π00 (T + K) = {0}, so that Weyl’s theorem does not hold for T + K.
Recall that T ∈ L(X) is said to be a Riesz operator if λI − T ∈ Φ(X)
for all λ 6= 0. Evidently, quasi-nilpotent operators and compact operators
are Riesz operators. A bounded operator T ∈ L(X) is said to be isoloid if
every isolated point of the spectrum is an eigenvalue. T ∈ L(X) is said to
be finite-isoloid if every isolated spectral point is an eigenvalue having finite
multiplicity.
A result of W. Y. Lee and S. H. Lee [25] shows that Weyl’s theorem for an
isoloid operator is also preserved by perturbations of commuting finite rank
operators. This result has been generalized by Oudghiri [29] as follows:
Theorem 2.7. If T ∈ L(X) is an isoloid operator which satisfies Weyl’s
theorem, if ST = T S, S ∈ L(X), and there exists n ∈ N such that S n is
finite-dimensional, then T + S satisfies Weyl’s theorem.
More recently, Y. M. Han and W. Y. Lee in [22] have shown that in the
case of Hilbert spaces if T is a finite-isoloid operator which satisfies Weyl’s
theorem and if S a compact operator commuting with T then also T + S
satisfies Weyl’s theorem. Again, Oudghiri [29] has shown that we have much
more:
Theorem 2.8. If T ∈ L(X) is a finite-isoloid operator which satisfies Weyl’s
theorem and if K is a Riesz operator commuting with T then also T + K
satisfies Weyl’s theorem.
Recall that a bounded operator T is said to be algebraic if there exists
a non-trivial polynomial h such that h(T ) = 0. The following two results
show that Weyl’s theorem survives under commuting algebraic perturbations
in some special cases.
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Weyl’s type theorems and perturbations
65
Theorem 2.9. [29] Suppose that T ∈ L(X) has property H(p), K algebraic
and T K = KT . Then Weyl’s theorem holds for T + K
As observed in (G) the property H(p) may fail for paranormal operators,
in particular fails for quasi-hyponormal operators [6]. However, since quasihyponormal operators are paranormal we can apply next result to this case.
Theorem 2.10. [6] Suppose that T ∈ L(H) is paranormal, K algebraic and
T K = KT . Then Weyl’s theorem holds for T + K.
It is well-known that if S ∈ L(X), and there exists n ∈ N such that S n
is finite-dimensional then S algebraic. If T satisfies H(p), or is paranormal,
then T is isoloid (see [2]), so that the result of Theorem 2.7 in these special
cases also follows from Theorem 2.9 and Theorem 2.10.
In the perturbation theory the ”commutative” condition is rather rigid. On
the other hand, it is known that without the commutativity, the spectrum can
however undergo a large change under even rank one perturbations. However,
we have the following result due to Y. M. Han and W. Y. Lee [22].
Theorem 2.11. Suppose that T ∈ L(H) is a finite-isoloid operator which
satisfies Weyl’s theorem. If σ(T ) has no holes (bounded components of the
complement) and has at most finitely many isolated points then Weyl’s theorem holds for T + K, where K ∈ L(H) is either a compact or quasi-nilpotent
operator commuting with T modulo the compact operators.
The result below follows immediately from Theorem 2.11
Corollary 2.12. Suppose that T ∈ L(H) satisfies Weyl’s theorem. If σ(T )
has no holes and has at most finitely many isolated points then Weyl’s theorem
holds for T + K for every compact operator K.
Corollary 2.12 applies to Toeplitz operators and not quasi-nilpotent unilateral weighted shifts, see [22].
3
a-Weyl’s theorem and perturbations
For a bounded operator T ∈ L(X) on a Banach space X let us denote
pa00 (T ) := σa (T ) \ σub (T ) = {λ ∈ σa (T ) : λI − T ∈ B+ (X)}.
We have,
a
pa00 (T ) ⊆ π00
(T ) for every T ∈ L(X).
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
In fact, if λ ∈ pa00 (T ) then is λI − T ∈ Φ+ (X) and p(λI − T ) < ∞. By
Remark 1.2 then λ is isolated in σa (T ). Furthermore, 0 < α(λI − T ) < ∞
since (λI − T )(X) is closed and λ ∈ σa (T ).
A bounded operator T ∈ L(X) is said to satisfy a-Browder’s theorem if
σwa (T ) = σub (T ).
Also a-Browder’s theorem may be characterized by localized SVEP:
T satisfies a-Browder’s theorem ⇔ T has SVEP at all λ ∈
/ σwa (T ).
An approximate point variant of Weyl’s theorem is a-Weyl’s theorem: according to Rakočević [32] an operator T ∈ L(X) is said to satisfy a-Weyl’s
theorem if
a
∆a (T ) := σa (T ) \ σwa (T ) = π00
(T ).
It should be noted that the set ∆a (T ) may be empty. This is, for instance,
the case of a right shift on ℓ2 (N), see [4]. Furthermore,
a-Weyl’s theorem holds for T ⇒ Weyl’s theorem holds for T,
while the converse does not hold, in general. Note that a-Weyl’s theorem
entails also a-Browder’s theorem. Precisely, we have:
Theorem 3.1. [2] For T ∈ L(X) the following statements are equivalent:
(i) T satisfies a-Weyl’s theorem;
a
(T ).
(ii) T satisfies a-Browder’s theorem and pa00 (T ) = π00
The following two results show that a-Weyl’s theorem is satisfied by a
considerable number of operators. In the sequel we shall denote by H(σ(T ))
the set of all analytic functions defined on a neighborhood of σ(T ). If T ∈
L(X) then for every f ∈ H(σ(T )) the operator f (T ) is defined by the classical
functional calculus.
Theorem 3.2. [2] If T ∈ L(X) has property (Hp ) then a-Weyl’s holds for
f (T ∗ ) for every f ∈ H(σ(T )). Analogously, if T ∗ has property (Hp ) then
a-Weyl’s holds for f (T ) for every f ∈ H(σ(T )).
Theorem 3.3. [2] If T ′ ∈ L(H) has property (Hp ) or T ′ is paranormal, then
a-Weyls theorem holds for f (T ) for every f ∈ H(σ(T )).
It is easy to find an example of an operator such that a-Weyl’s theorem
holds for T while there is a commuting finite rank operator K such that
a-Weyl’s theorem fails for T + K.
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Weyl’s type theorems and perturbations
67
Example 3.4. Let Q be any injective quasi-nilpotent operator on a Banach
space X. Define T := Q ⊕ I on X ⊕ X. Clearly, T satisfies a-Weyl’s theorem.
Take P ∈ L(X) any finite rank projection and set K := 0⊕(−P ). Then T K =
a
KT and 0 ∈ π00
(T + K) ∩ σwa (T ∗ K). Clearly, 0 ∈
/ σa (T + K) \ σwa (T + K),
a
so that σa (T + K) \ σwa (T + k) 6= π00
(T ) and hence a-Weyl’s theorem does
not holds for T + K.
To see when a-Weyls theorem is transmitted under perturbation we need
to introduce some definitions. A bounded operator T ∈ L(X) is said to be
a-isoloid if every isolated point of the approximate point spectrum σa (T ) is
an eigenvalue. T ∈ L(X) is said to be finite a-isoloid if every isolated point
σa (T ) is an eigenvalue having finite multiplicity. The following two results are
due to Oudghiri [30].
Theorem 3.5. [30] If T ∈ L(X) is an a-isoloid operator which satisfies aWeyl’s theorem, if ST = T S, S ∈ L(X), and there exists n ∈ N such that S n
is finite-dimensional, then T + S satisfies a-Weyl’s theorem.
Theorem 3.5 extends a result of D. S. Djordjević [19], see also Theorem
2.3 of [12], where a-Weyl’s theorem was proved for T + S when S is a finite
rank operator commuting with T .
Theorem 3.6. [30] If T ∈ L(X) is a finite a-isoloid operator which satisfies
a-Weyl’s theorem and if K a Riesz operator commuting with T then also T +K
satisfies a-Weyl’s theorem.
In particular, Theorem 3.6 applies to compact perturbations T + K.
Theorem 3.7. [12] Suppose that T ∈ L(X) and Q is an injective quasinilpotent operator commuting with T . If T satisfies a-Weyl’s theorem then
also T + Q satisfies a-Weyl’s theorem.
It is easily seen that quasi-nilpotent operators do not satisfy a-Weyl’s
theorem, in general. For instance, if
T (x1 , x2 , . . . , ) := (0,
x2 x3
, , . . . ),
2 3
(xn ) ∈ ℓ2 (N)
then T is quasi-nilpotent but a-Weyl’s theorem fails for T .
Theorem 3.8. [12] Suppose that Q is quasi-nilpotent operator satisfying aWeyl’s theorem. If K is a finite rank operator commuting with Q then Q + K
satisfies a-Weyl’s theorem.
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4
Pietro Aiena, Jesús R. Guillen, Pedro Peña
Property (ω) and perturbations
Weyl’s theorem admits another interesting variant, introduced by Rakočević
in [31] and studied in the recent paper [10].
Definition 4.1. A bounded operator T ∈ L(X) satisfies property (w) if
σa (T ) \ σwa (T ) = π00 (T ).
As observed in [10], we also have:
T satisfies property (w) ⇒ Weyl’s theorem holds for T,
(12)
and examples of operators satisfying Weyl’s theorem but not property (w)
may be found in [10]. Property (w) is fulfilled by a relevant number of Hilbert
space operators, see [10], and this property is equivalent to Weyl’s theorem for
T whenever T ∗ satisfies SVEP ([10, Theorem 2.16]). For instance, property
(w) is satisfied by generalized scalar operator, or if the Hilbert adjoint T ′ has
property H(p) [10, Corollary 2.20]. Note that
T satisfies property (w) ⇒ a-Browder’s theorem holds for T,
and precisely we have:
Theorem 4.2. [10] If T ∈ L(X) then the following statements are equivalent:
(i) T satisfies property (w);
(ii) a-Browder’s theorem holds for T and pa00 (T ) = π00 (T ).
Note that property (w) is not intermediate between Weyl’s theorem and
a-Weyl’s theorem, see [10] for examples. Property (w) is preserved by commuting finite-dimensional perturbations in the case that T is a-isoloid.
Theorem 4.3. [3] Suppose that T ∈ L(X) is a-isoloid and satisfies property
(w), K is a finite rank operator that commutes with T . Then T + K satisfies
property (w).
A similar result holds for nilpotent perturbations.
Theorem 4.4. [3] Suppose that T ∈ L(X) is a-isoloid. If T satisfies property
(w) and N is nilpotent operator that commutes with T then T + N satisfies
property (w).
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
Weyl’s type theorems and perturbations
69
Example 4.5. Both Theorem 4.4 and Theorem 4.3 fail if we assume that
the nilpotent operator N , and the finite rank operator K do not commute
with T . For instance, let T and N be as in Example 2.5. Then T satisfies
Weyl’s theorem and is decomposable, since T is a quasi-nilpotent operator.
This implies that T satisfies property (w), by Corollary 2.10 of [10].
On the other hand, T + N does not satisfy Weyl’s theorem, and consequently, by the implication (12), T + N does not satisfy property (w).
Example 4.6. The following example shows that Theorem 4.3 fails if we do
not assume that T is a-isoloid. Let T and K be defined as in Example 2.6.
Clearly, α(T ) = 0 so T is not a-isoloid. Now, T satisfies Weyl’s theorem, and
hence by Theorem 2.16 of [10] T satisfies property (w).
On the other hand, Weyl’s theorem does not hold for T + K and this
implies (see (12)) that property (w) fails for T + K.
Example 4.7. In general, property (w) is not transmitted from T to a quasinilpotent perturbation T + Q. For instance, take T = 0, and Q ∈ L(ℓ2 (N))
defined by
x2 x3
Q(x1 , x2 , . . . ) = ( , , . . . ) for all (xn ) ∈ ℓ2 (N,
2 3
Then Q is quasi-nilpotent and {0} = π00 (Q) 6= σa (Q) \ σaw (Q) = ∅ . Hence
T satisfies property (w) but T + Q = Q fails this property.
However, property (w) is preserved under injective quasi-nilpotent perturbations.
Theorem 4.8. [3] Suppose that T ∈ L(X) and Q is an injective quasinilpotent operator commuting with T . If T satisfies property (w) then also
T + Q satisfies property (w).
Property (w) is preserved under finite rank perturbations or nilpotent
perturbations commuting with T in the case that T ∗ ∈ L(X) has property
H(p).
Theorem 4.9. [3] Suppose that T ∗ ∈ L(X) has property H(p), K is either a
finite rank operator or a nilpotent operator commuting with T . Then property
(w) holds for T + K.
An analogous result holds for paranormal operators acting on Hilbert
spaces.
Theorem 4.10. [3] Suppose that T ′ ∈ L(H) is paranormal and K is either a
finite rank operator or a nilpotent operator commuting with T . Then property
(w) holds for T + K.
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70
Pietro Aiena, Jesús R. Guillen, Pedro Peña
Recall that every generalized scalar operator satisfies property (w) [10,
Corollary 2.11].
Theorem 4.11. [3] Suppose that T ∈ L(X) is generalized scalar operator, K
an algebraic operator commuting with T . Then property (w) holds for T + K.
References
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of the approximate point spectrum, J. Math. Anal. Appl. 279(1)(2003),
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[13] L. Chen, R. Yingbin, Y. Zikun w-hyponormal operators are subscalar.,
Int. Eq. Oper. Theory 50(2) 165-168.
[14] N. N. Chourasia, P. B. Ramanujan Paranormal operators on Banach
spaces Bull. Austral. Math. Soc. 21(1980), 161-168.
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in Mathematics 51(1981).
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Math. Debrecen 55, 3-4(3)(1999), 283-298.
[20] J. K. Finch The single valued extension property on a Banach space Pacific J. Math. 58(1975), 61-69.
[21] Y. M. Han, W. Y. Lee Weyl’s theorem holds for algebraically hyponormal
operators. Proc. Amer. Math. Soc. 128(8)(2000), 2291-2296.
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148(2001), 193-206.
[23] Y. M. Han, A. H. Kim A note on ∗-paranormal operators Int. Eq. Oper.
Theory. 49(2004), 435-444.
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Clarendon Press, Oxford, 2000.
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549-553.
[26] M. Mbekhta Sur la théorie spectrale locale et limite des nilpotents. Proc.
Amer. Math. Soc. 110 (1990), 621-631.
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SVEP. Studia Math. 163(1)(2004), 85-101.
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[29] M. Oudghiri Weyl’s theorem and perturbations. (2005) To appear in
Integr. Equ. Oper. Theory.
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[31] V. Rakočević On a class of operators Mat. Vesnik 37(1985), 423-426.
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[33] V. Rakočević Approximate point spectrum and commuting compact perturbations Glasgow Math. J. 28(1986), 193-198.
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Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
Weyl’s type theorems and
perturbations
Teoremas de tipo Weyl y perturbaciones
Pietro Aiena ([email protected])
Dipartimento di Matematica ed Applicazioni
Facoltà di Ingegneria, Università di Palermo
Viale delle Scienze, I-90128 Palermo (Italy)
Jesús R. Guillen ([email protected])
Departamento de Matemáticas, Facultad de Ciencias
ULA; Merida (Venezuela)
Pedro Peña ([email protected])
Departamento de Matemáticas, Facultad de Ciencias
ULA; Merida (Venezuela)
Abstract
Weyl’s theorem for a bounded linear operator T on complex Banach
spaces, as well as its variants, a-Weyl’s theorem and property (w), in
general is not transmitted to the perturbation T + K, even when K
is a ”good” operator, as a commuting finite rank operator or compact
operator. Weyl’s theorems do not survive also if K is a commuting
quasi-nilpotent operator. In this paper we discuss some sufficient conditions for which Weyl’s theorem, a-Weyl’s theorem as well as property
(w) is transmitted under such kinds of perturbations.
Key words and phrases: Local spectral theory, Fredholm theory,
Weyl’s theorem.
Resumen
El Teorema de Weyl para un operador lineal acotado T sobre un
espacio de Banach complejo, como también sus variantes, el teorema
a-Weyl y la propiedad (w), en general no se transmite a la perturbación
T + K, ni aún cuando K es un ”buen” operador, como operadores
Received 2006/03/12. Revised 2006/07/02. Accepted 2006/07/22.
MSC Primary 47A10, 47A11. Secondary 47A53, 47A55.
56
Pietro Aiena, Jesús R. Guillen, Pedro Peña
conmutantes de rango finito u operadores compactos. Los teoremas de
Weyl no sobreviven tampoco si K es un operador conmutante cuasinilpotente. En este artı́culo discutimos algunas condiciones suficientes
para las cuales el teorema de Weyl, el teorema a-Weyl y la propiedad
(w) se transmite bajo tales tipos de perturbaciones.
Palabras y frases clave: Teorı́a espectral local, teorı́a de Fredholm,
teorema de Weyl.
1
Introduction and preliminaries
Weyl [35] examined the spectra of all compact perturbations of a hermitian
operator T on a Hilbert space and proved that their intersection coincides
with the isolated point of the spectrum σ(T ) which are eigenvalues of finite
multiplicity. Weyl’s theorem has been extended to several classes of Hilbert
spaces operators and Banach spaces operators. More recently, two variants
of Weyl’s theorem have been introduced by Rakočević [32], [31], the so called
a-Weyl’s theorem and the property (w) studied also in [10].
In general, Weyl’s theorem is not sufficient for Weyl’s theorem for T + K,
where K ∈ L(X). Weyl’s theorems are liable to fail also under ”small” perturbations if ”small” is interpreted in the sense of compact or quasi-nilpotent
operators. In this note, we study sufficient conditions for which we have the
stability of Weyl’s theorems and property (w), under perturbations by finite
rank operators, compact operators, or quasi-nilpotent operator commuting
with T . We shall see that the stability of Weyl’s theorems requires some
special conditions on the isolated points of the spectrum (or on the isolated
points of the approximate point spectrum).
We begin with some standard notations on Fredholm theory. Throughout
this note, let L(X) denote the algebra of all bounded linear operators acting
on an infinite dimensional complex Banach space X. If T ∈ L(X) write α(T )
for the dimension of the kernel ker T and β(T ) for the codimension of the
range T (X). Denote by
Φ+ (X) := {T ∈ L(X) : α(T ) < ∞ and T (X) is closed}
the class of all upper semi-Fredholm operators, and by
Φ− (X) := {T ∈ L(X) : β(T ) < ∞}
the class of all lower semi-Fredholm operators. The class of all semi-Fredholm
operators is defined by Φ± (X) := Φ+ (X) ∪ Φ− (X), while the class of all
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
Weyl’s type theorems and perturbations
57
Fredholm operators is defined by Φ(X) := Φ+ (X) ∩ Φ− (X). The index of
a semi-Fredholm operator is defined by ind T := α(T ) − β(T ). Recall that
the ascent p := p(T ) of a linear operator T is defined to be the smallest nonnegative integer p such that ker T p = ker T p+1 (X). If such an integer does
not exist we put p(T ) = ∞. Analogously, the descent q := q(T ) of an operator
T is the smallest non-negative integer q such that T q (X) = T q+1 (X), and if
such an integer does not exist we put q(T ) = ∞. It is well-known that if p(T )
and q(T ) are both finite then p(T ) = q(T ), [1, Theorem 3.3]. Two important
classes of operators are the class of all upper semi-Browder operators
B+ (X) := {T ∈ Φ+ (X) : p(T ) < ∞},
and the class of all lower semi-Browder operators
B− (X) := {T ∈ Φ− (X) : q(T ) < ∞}.
The class of all Browder operators is defined by B(X) := B+ (X) ∩ B− (X).
Recall that a bounded operator T ∈ L(X) is said to be a Weyl operator if
T ∈ Φ(X) and ind T = 0. Clearly, if T is Browder then T is Weyl, since the
finiteness of p(T ) and q(T ) implies, for a Fredholm operator, that T has index
0, see [1, Theorem 3.4].
These classes of operators motivate the definition of several spectra. The
upper semi-Browder spectrum of T ∈ L(X) is defined by
σub (T ) := {λ ∈ C : λI − T ∈
/ B+ (X)},
the lower semi-Browder spectrum of T ∈ L(X) is defined by
σlb (T ) := {λ ∈ C : λI − T ∈
/ B− (X)},
while the Browder spectrum of T ∈ L(X) is defined by
σb (T ) := {λ ∈ C : λI − T ∈
/ B(X)}.
The Weyl spectrum of T ∈ L(X) is defined by
σw (T ) := {λ ∈ C : λI − T is not Weyl}.
We have that σw (T ) = σw (T ∗ ), where T ∗ denotes the dual of T , while
σub (T ) = σlb (T ∗ ),
σlb (T ) = σub (T ∗ ).
Evidently,
σw (T ) ⊆ σb (T ) = σw (T ) ∪ acc σ(T ),
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
where we write acc K for the accumulation points of K ⊆ C, see [1, Chapter
3]. The Weyl (or essential) approximate point spectrum σwa (T ) of a bounded
operator T ∈ L(X) is the complement of those λ ∈ C for which λI − T ∈
Φ+ (X) and ind (λI − T ) ≤ 0. Note that σwa (T ) is the intersection of all
approximate point spectra σa (T + K) of compact perturbations K of T , see
[32]. The Weyl surjectivity spectrum σws (T ) is the complement of those λ ∈ C
for which λI − T ∈ Φ− (X) and ind (λI − T ) ≥ 0. The spectrum σwa (T )
coincides with the intersection of all surjectivity spectra σs (T +K) of compact
perturbations K of T , see [32] or [1, p.151]. Clearly, the last two spectra are
dual each other, i.e.,
σwa (T ) = σws (T ⋆ ) and
σws (T ) = σwa (T ⋆ ).
Furthermore, σw (T ) = σwa (T )∪σws (T ). Since p(T ) < ∞ (respectively, q(T ) <
∞) entails that ind T ≤ 0 (respectively, ind T ≥ 0), we have σwa (T ) ⊆ σub (T )
and σws (T ) ⊆ σlb (T ), and the precise relationship between these spectra is
given by the following equalities:
σub (T ) = σwa (T ) ∪ acc σa (T ),
(1)
σlb (T ) = σws (T ) ∪ acc σs (T ),
(2)
and
see [33]. This article also deals with the single valued extension property. This
property has a basic role in the local spectral theory, see the recent monograph
of Laursen and Neumann [24] or Aiena [1]. In this article we shall consider a
localized version of this property, recently studied by several authors [8], [5],
[9], and previously by Finch [20], and Mbekhta [26].
Definition 1.1. Let X be a complex Banach space and T ∈ L(X). The
operator T is said to have the single valued extension property at λ0 ∈ C
(abbreviated SVEP at λ0 ), if for every open disc U of λ0 , and f : U → X
satisfying the equation (λI − T )f (λ) = 0 for all λ ∈ U we have f ≡ 0 on U .
An operator T ∈ L(X) is said to have the SVEP if T has the SVEP at every
point λ ∈ C.
An operator T ∈ L(X) has the SVEP at every point of the resolvent
ρ(T ) := C \ σ(T ). The identity theorem for analytic function ensures that
for every T ∈ L(X), both T and T ∗ have the SVEP at the points of the
boundary ∂σ(T ) of the spectrum σ(T ). In particular, that both T and T ∗
have SVEP at every isolated point of σ(T ) = σ(T ∗ ). The SVEP is inherited
by the restrictions to closed invariant subspaces, i.e. if T ∈ L(X) has the
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Weyl’s type theorems and perturbations
59
SVEP at λ0 and M is a closed T -invariant subspace then T |M has SVEP at
λ0 .
Recall that T ∈ L(X) is said to be bounded below if T is injective and has
closed range. Let σa (T ) denote the classical approximate point spectrum of T ,
i. e. the set
σa (T ) := {λ ∈ C : λI − T is not bounded below},
and let
σs (T ) := {λ ∈ C : λI − T is not surjective}
denote the surjectivity spectrum of T . Note that
p(λI − T ) < ∞ ⇒ T has SVEP at λ,
(3)
q(λI − T ) < ∞ ⇒ T ∗ has SVEP at λ,
(4)
and dually
see [1, Theorem 3.8]. From the definition of SVEP we also have
σa (T ) does not cluster at λ ⇒ T has SVEP at λ,
(5)
σs (T ) does not cluster at λ ⇒ T ∗ has SVEP at λ.
(6)
and dually
The quasi-nilpotent part of T is defined by
1
H0 (T ) := {x ∈ X : lim kT n xk n = 0}.
n→∞
It is easily seen that H0 (T ) is a linear T -invariant subspace and ker (T m ) ⊆
H0 (T ) for every m ∈ N. The subspace H0 (T ) may be not closed and coincides
with the local spectral subspace XT ({0} in the case where T has SVEP, see
Theorem 2.20 of [1]. The following implication holds for every operator T ∈
L(X),
H0 (λI − T ) is closed ⇒ T has SVEP at λ,
(7)
see [5].
Remark 1.2. There is a fundamental relationship between SVEP and the Fredholm theory: all the implications (3)-(7) become equivalences if we assume
that λI − T ∈ Φ± (X), see [5], [9].
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
Weyl’s theorem under perturbations
If T ∈ L(X) let us consider the complement in σ(T ) of the Browder spectrum:
p00 (T ) := σ(T ) \ σb (T ) = {λ ∈ σ(T ) : λI − T is Browder}.
Write iso K for the set of all isolated points of K ⊆ C, and set
π00 (T ) := {λ ∈ iso σ(T ) : 0 < α(λI − T ) < ∞},
the set of isolated eigenvalues of finite multiplicities. Obviously,
p00 (T ) ⊆ π00 (T ) for every T ∈ L(X).
(8)
Following Coburn [15], we say that Weyl’s theorem holds for T ∈ L(X) if
∆(T ) := σ(T ) \ σw (T ) = π00 (T ),
(9)
while T is said to satisfy Browder’s theorem if
σ(T ) \ σw (T ) = p00 (T ),
or equivalently, σw (T ) = σb (T ). Note that
Weyl’s theorem ⇒ Browder’s theorem,
see, for instance [1, p. 166]. Browder’s theorem may be characterized by
means of SVEP:
T satisfies Browder’s theorem ⇔ T has SVEP at all λ ∈
/ σw (T ).
Browder’s theorem corresponds to the half of Weyl’s theorem, in the following sense:
Theorem 2.1. [2] If T ∈ L(X) then the following assertions are equivalent:
(i) Weyl’s theorem holds for T ;
(ii) T satisfies Browder’s theorem and π00 (T ) = p00 (T );
The conditions p00 (T ) = π00 (T ) is equivalent to several other conditions,
see [1, Theorem 3.84]. The equality p00 (T ) = π00 (T ) may be described in
another way. To see this, let P0 (X), where X is a Banach space, denote the
class of all operators T ∈ L(X) such that there exists p := p(λ) ∈ N for which
H0 (λI − T ) = ker (λI − T )p
for all λ ∈ π00 (T ).
Then we have
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
(10)
Weyl’s type theorems and perturbations
61
Theorem 2.2. [6] With the notation above, T ∈ P0 (X) if and only if p00 (T ) =
π00 (T ). In particular, if T has SVEP then Weyl’s theorem holds for T if and
only if T ∈ P0 (X).
The following classes of operators has been introduced by Oudghiri [28],
see also [11].
Definition 2.3. A bounded operator T ∈ L(X) is said to satisfy property
H(p) if
H0 (λI − T ) = ker (λI − T )p for all λ ∈ C.
(11)
for some p = p(λ) ∈ N, .
From the implication in (7) it is clear that property H(p) entails SVEP.
Many of the commonly considered operators on Banach spaces and Hilbert
spaces have property H(p), or have SVEP and belong to the class P0 (X). We
list now some classes of operators for which property H(p) holds. In the sequel
by X we shall denote a Banach space, while a Hilbert space will be denoted
by H.
(A) A bounded operator T ∈ L(X) is said to be paranormal if
kT xk2 ≤ kT 2 xkkxk for all x ∈ X.
The class of paranormal operators properly contains a relevant number of
Hilbert space operators, among them p-hyponormal operators, log-hyponormal
operators, quasi-hyponormal operators (see later for definitions). Every paranormal operator T ∈ L(H) has SVEP. This may be easily seen as follows: if
λ 6= 0 and λ 6= µ then, by Theorem 2.6 of [14], we have kx + yk ≥ kyk whenever x ∈ ker (µI − T ) and y ∈ ker (λI − T ). It then follows that if U is an open
disc and f : U → X is an analyitic function such that 0 6= f (z) ∈ ker (zI − T )
for all z ∈ U , then f fails to be continuous at every 0 6= λ ∈ U .
(B) An operator T ∈ L(X) is called totally paranormal if λI − T is paranormal for all λ ∈ C. Every totally paranormal operator satisfies property
H(1), see [11]. In the sequel we shall denote by T ′ the Hilbert adjoint of
T ∈ L(H).
An operator T ∈ L(H) is said to be *-paranormal if
kT ′ xk2 ≤ kT 2 xk
holds for all unit vectors x ∈ H. T ∈ L(H) is said to be totally *-paranormal
if λI − T is *-paranormal for all λ ∈ C. Every totally ∗-paranormal operator
satisfies property H(1), see [23]. The class of totally paranormal operators includes all hyponormal operators on Hilbert spaces H. Recall that the operator
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
T ∈ L(H) is said to be hyponormal if
kT ′ xk ≤ kT xk
for all x ∈ X.
Hyponormal operators are p-hyponormal with p = 1, where T ∈ L(H) is said
to be p-hyponormal, with 0 < p ≤ 1, if
(T ′ T )p ≥ (T T ′ )p .
Every injective p-hyponormal operator satisfies property H(1), see [11]. The
concept of hyponormality may be relaxed as follows: T ∈ L(H) is said to be
quasi-hyponormal if
2
(T ′ T )2 ≤ T ′ T 2
Also quasi-normal operators are totally paranormal, since these operators are
hyponormal, see Conway [16] for details.
(C) An operator T ∈ L(H) is said to be log-hyponormal if T is invertible
and satisfies
log (T ′ T ) ≥ log (T T ′ ).
Every log-hyponormal operator satisfies the condition H(1), see [11]. In fact,
log-hyponormal operator are similar to hyponormal operators and property
H(1) is preserved by similarity [11, Theorem 2.4].
(D) Let A be a commutative Banach algebra. A linear map T : A → A
is said to be a multiplier if (T x)y = x(T y) holds for all x, y ∈ A. If A is a
commutative semi-simple Banach algebra, then every multiplier satisfies the
condition H(1) see [5]. Moreover, every multiplier of a commutative semisimple Banach algebra has SVEP, [1, Chapter 4]. In particular, the condition
H(1) holds for every convolution operator on the group algebra L1 (G), where
G is a locally compact abelian group.
(E) An important class of operators which satisfy property H(p) is given
by the class of subscalar operators [28]. Recall that T ∈ L(X) is said to
be generalized scalar if there exists a continuous algebra homomorphism Ψ :
C ∞ (C) → L(X) such that
Ψ(1) = I
and Ψ(Z) = T,
where C ∞ (C) denote the Fréchet algebra of all infinitely differentiable complexvalued functions on C, and Z denotes the identity function on C. Every generalized scalar operator has SVEP, see [24]. An operator is subscalar whenever
is similar to the restriction of a generalized scalar operator to one of its closed
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Weyl’s type theorems and perturbations
63
invariant subspaces. The interested reader can find a well organized study of
these operators in the book by Laursen and Neumann [24]. Property H(p)
is then satisfied by p-hyponormal operators and log-hyponormal operators,
M -hyponormal operators [24, Proposition 2.4.9], and totally paranormal operators [11], since all these operators are subscalar.
(F) An operator T ∈ L(X) for which there exists a complex non-constant
polynomial h such that h(T ) is paranormal is said to be algebraically paranormal. If T ∈ L(H) is algebraically paranormal then T ∈ P0 (H), so it satisfies
the condition (10), see [2], but in general the condition H(p) is not satisfied
by paranormal operators, (for an example see [6, Example 2.3]). Since every paranormal operator has SVEP then also every algebraically paranormal
operator has SVEP, see Theorem 2.40 of [1] or [18], so that Weyl’s theorem
holds for every algebraically paranormal operator, see also [21].
Weyl’s theorem is transmitted by some special perturbations. The following result has been first proved by Oberai [27].
Theorem 2.4. Weyl’s theorem is transmitted from T ∈ L(X) to T + N when
N is a nilpotent operator commuting with T .
The following example shows that Oberai’s result does not hold if we do
not assume that the nilpotent operator N commutes with T
Example 2.5. Let X := ℓ2 (N) and T and N be defined by
T (x1 , x2 , . . . , ) := (0,
x1 x2
, , . . . ),
2 3
(xn ) ∈ ℓ2 (N)
and
x1
, 0, 0, . . . ), (xn ) ∈ ℓ2 (N)
2
Clearly, N is a nilpotent operator, and T is a quasi-nilpotent operator satisfying Weyl’s theorem. On the other hand, it is easily seen that 0 ∈ π00 (T + N )
and 0 ∈
/ σ(T +N )\σw (T +N ), so that T +N does not satisfies Weyl’s theorem.
N (x1 , x2 , . . . , ) := (0, −
Note that the operator N in Example 2.5 is also a finite rank operator not
commuting with T . In general, Weyl’s theorem is also not transmitted under
commuting finite rank perturbation.
Example 2.6. Let S : ℓ2 (N) → ℓ2 (N) be an injective quasi-nilpotent operator, and let U : ℓ2 (N) → ℓ2 (N) be defined :
U (x1 , x2 , . . . , ) := (−x1 , 0, 0, . . . ),
with (xn ) ∈ ℓ2 (N).
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
Define on X := ℓ2 (N) ⊕ ℓ2 (N) the operators T and K by
T := I ⊕ S
and K := U ⊕ 0
Clearly, K is a finite rank operator and KT = T K. It is easy to check that
σ(T ) = σw (T ) = σa (T ) = {0, 1}.
Now, both T and T ∗ have SVEP, since σ(T ) = σ(T ∗ ) is finite. Moreover,
π00 (T ) = σ(T ) \ σw (T ) = ∅, so T satisfies Weyl’s theorem.
On the other hand,
σ(T + K) = σw (T + K) = {0, 1},
and π00 (T + K) = {0}, so that Weyl’s theorem does not hold for T + K.
Recall that T ∈ L(X) is said to be a Riesz operator if λI − T ∈ Φ(X)
for all λ 6= 0. Evidently, quasi-nilpotent operators and compact operators
are Riesz operators. A bounded operator T ∈ L(X) is said to be isoloid if
every isolated point of the spectrum is an eigenvalue. T ∈ L(X) is said to
be finite-isoloid if every isolated spectral point is an eigenvalue having finite
multiplicity.
A result of W. Y. Lee and S. H. Lee [25] shows that Weyl’s theorem for an
isoloid operator is also preserved by perturbations of commuting finite rank
operators. This result has been generalized by Oudghiri [29] as follows:
Theorem 2.7. If T ∈ L(X) is an isoloid operator which satisfies Weyl’s
theorem, if ST = T S, S ∈ L(X), and there exists n ∈ N such that S n is
finite-dimensional, then T + S satisfies Weyl’s theorem.
More recently, Y. M. Han and W. Y. Lee in [22] have shown that in the
case of Hilbert spaces if T is a finite-isoloid operator which satisfies Weyl’s
theorem and if S a compact operator commuting with T then also T + S
satisfies Weyl’s theorem. Again, Oudghiri [29] has shown that we have much
more:
Theorem 2.8. If T ∈ L(X) is a finite-isoloid operator which satisfies Weyl’s
theorem and if K is a Riesz operator commuting with T then also T + K
satisfies Weyl’s theorem.
Recall that a bounded operator T is said to be algebraic if there exists
a non-trivial polynomial h such that h(T ) = 0. The following two results
show that Weyl’s theorem survives under commuting algebraic perturbations
in some special cases.
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Weyl’s type theorems and perturbations
65
Theorem 2.9. [29] Suppose that T ∈ L(X) has property H(p), K algebraic
and T K = KT . Then Weyl’s theorem holds for T + K
As observed in (G) the property H(p) may fail for paranormal operators,
in particular fails for quasi-hyponormal operators [6]. However, since quasihyponormal operators are paranormal we can apply next result to this case.
Theorem 2.10. [6] Suppose that T ∈ L(H) is paranormal, K algebraic and
T K = KT . Then Weyl’s theorem holds for T + K.
It is well-known that if S ∈ L(X), and there exists n ∈ N such that S n
is finite-dimensional then S algebraic. If T satisfies H(p), or is paranormal,
then T is isoloid (see [2]), so that the result of Theorem 2.7 in these special
cases also follows from Theorem 2.9 and Theorem 2.10.
In the perturbation theory the ”commutative” condition is rather rigid. On
the other hand, it is known that without the commutativity, the spectrum can
however undergo a large change under even rank one perturbations. However,
we have the following result due to Y. M. Han and W. Y. Lee [22].
Theorem 2.11. Suppose that T ∈ L(H) is a finite-isoloid operator which
satisfies Weyl’s theorem. If σ(T ) has no holes (bounded components of the
complement) and has at most finitely many isolated points then Weyl’s theorem holds for T + K, where K ∈ L(H) is either a compact or quasi-nilpotent
operator commuting with T modulo the compact operators.
The result below follows immediately from Theorem 2.11
Corollary 2.12. Suppose that T ∈ L(H) satisfies Weyl’s theorem. If σ(T )
has no holes and has at most finitely many isolated points then Weyl’s theorem
holds for T + K for every compact operator K.
Corollary 2.12 applies to Toeplitz operators and not quasi-nilpotent unilateral weighted shifts, see [22].
3
a-Weyl’s theorem and perturbations
For a bounded operator T ∈ L(X) on a Banach space X let us denote
pa00 (T ) := σa (T ) \ σub (T ) = {λ ∈ σa (T ) : λI − T ∈ B+ (X)}.
We have,
a
pa00 (T ) ⊆ π00
(T ) for every T ∈ L(X).
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
In fact, if λ ∈ pa00 (T ) then is λI − T ∈ Φ+ (X) and p(λI − T ) < ∞. By
Remark 1.2 then λ is isolated in σa (T ). Furthermore, 0 < α(λI − T ) < ∞
since (λI − T )(X) is closed and λ ∈ σa (T ).
A bounded operator T ∈ L(X) is said to satisfy a-Browder’s theorem if
σwa (T ) = σub (T ).
Also a-Browder’s theorem may be characterized by localized SVEP:
T satisfies a-Browder’s theorem ⇔ T has SVEP at all λ ∈
/ σwa (T ).
An approximate point variant of Weyl’s theorem is a-Weyl’s theorem: according to Rakočević [32] an operator T ∈ L(X) is said to satisfy a-Weyl’s
theorem if
a
∆a (T ) := σa (T ) \ σwa (T ) = π00
(T ).
It should be noted that the set ∆a (T ) may be empty. This is, for instance,
the case of a right shift on ℓ2 (N), see [4]. Furthermore,
a-Weyl’s theorem holds for T ⇒ Weyl’s theorem holds for T,
while the converse does not hold, in general. Note that a-Weyl’s theorem
entails also a-Browder’s theorem. Precisely, we have:
Theorem 3.1. [2] For T ∈ L(X) the following statements are equivalent:
(i) T satisfies a-Weyl’s theorem;
a
(T ).
(ii) T satisfies a-Browder’s theorem and pa00 (T ) = π00
The following two results show that a-Weyl’s theorem is satisfied by a
considerable number of operators. In the sequel we shall denote by H(σ(T ))
the set of all analytic functions defined on a neighborhood of σ(T ). If T ∈
L(X) then for every f ∈ H(σ(T )) the operator f (T ) is defined by the classical
functional calculus.
Theorem 3.2. [2] If T ∈ L(X) has property (Hp ) then a-Weyl’s holds for
f (T ∗ ) for every f ∈ H(σ(T )). Analogously, if T ∗ has property (Hp ) then
a-Weyl’s holds for f (T ) for every f ∈ H(σ(T )).
Theorem 3.3. [2] If T ′ ∈ L(H) has property (Hp ) or T ′ is paranormal, then
a-Weyls theorem holds for f (T ) for every f ∈ H(σ(T )).
It is easy to find an example of an operator such that a-Weyl’s theorem
holds for T while there is a commuting finite rank operator K such that
a-Weyl’s theorem fails for T + K.
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Weyl’s type theorems and perturbations
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Example 3.4. Let Q be any injective quasi-nilpotent operator on a Banach
space X. Define T := Q ⊕ I on X ⊕ X. Clearly, T satisfies a-Weyl’s theorem.
Take P ∈ L(X) any finite rank projection and set K := 0⊕(−P ). Then T K =
a
KT and 0 ∈ π00
(T + K) ∩ σwa (T ∗ K). Clearly, 0 ∈
/ σa (T + K) \ σwa (T + K),
a
so that σa (T + K) \ σwa (T + k) 6= π00
(T ) and hence a-Weyl’s theorem does
not holds for T + K.
To see when a-Weyls theorem is transmitted under perturbation we need
to introduce some definitions. A bounded operator T ∈ L(X) is said to be
a-isoloid if every isolated point of the approximate point spectrum σa (T ) is
an eigenvalue. T ∈ L(X) is said to be finite a-isoloid if every isolated point
σa (T ) is an eigenvalue having finite multiplicity. The following two results are
due to Oudghiri [30].
Theorem 3.5. [30] If T ∈ L(X) is an a-isoloid operator which satisfies aWeyl’s theorem, if ST = T S, S ∈ L(X), and there exists n ∈ N such that S n
is finite-dimensional, then T + S satisfies a-Weyl’s theorem.
Theorem 3.5 extends a result of D. S. Djordjević [19], see also Theorem
2.3 of [12], where a-Weyl’s theorem was proved for T + S when S is a finite
rank operator commuting with T .
Theorem 3.6. [30] If T ∈ L(X) is a finite a-isoloid operator which satisfies
a-Weyl’s theorem and if K a Riesz operator commuting with T then also T +K
satisfies a-Weyl’s theorem.
In particular, Theorem 3.6 applies to compact perturbations T + K.
Theorem 3.7. [12] Suppose that T ∈ L(X) and Q is an injective quasinilpotent operator commuting with T . If T satisfies a-Weyl’s theorem then
also T + Q satisfies a-Weyl’s theorem.
It is easily seen that quasi-nilpotent operators do not satisfy a-Weyl’s
theorem, in general. For instance, if
T (x1 , x2 , . . . , ) := (0,
x2 x3
, , . . . ),
2 3
(xn ) ∈ ℓ2 (N)
then T is quasi-nilpotent but a-Weyl’s theorem fails for T .
Theorem 3.8. [12] Suppose that Q is quasi-nilpotent operator satisfying aWeyl’s theorem. If K is a finite rank operator commuting with Q then Q + K
satisfies a-Weyl’s theorem.
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
Property (ω) and perturbations
Weyl’s theorem admits another interesting variant, introduced by Rakočević
in [31] and studied in the recent paper [10].
Definition 4.1. A bounded operator T ∈ L(X) satisfies property (w) if
σa (T ) \ σwa (T ) = π00 (T ).
As observed in [10], we also have:
T satisfies property (w) ⇒ Weyl’s theorem holds for T,
(12)
and examples of operators satisfying Weyl’s theorem but not property (w)
may be found in [10]. Property (w) is fulfilled by a relevant number of Hilbert
space operators, see [10], and this property is equivalent to Weyl’s theorem for
T whenever T ∗ satisfies SVEP ([10, Theorem 2.16]). For instance, property
(w) is satisfied by generalized scalar operator, or if the Hilbert adjoint T ′ has
property H(p) [10, Corollary 2.20]. Note that
T satisfies property (w) ⇒ a-Browder’s theorem holds for T,
and precisely we have:
Theorem 4.2. [10] If T ∈ L(X) then the following statements are equivalent:
(i) T satisfies property (w);
(ii) a-Browder’s theorem holds for T and pa00 (T ) = π00 (T ).
Note that property (w) is not intermediate between Weyl’s theorem and
a-Weyl’s theorem, see [10] for examples. Property (w) is preserved by commuting finite-dimensional perturbations in the case that T is a-isoloid.
Theorem 4.3. [3] Suppose that T ∈ L(X) is a-isoloid and satisfies property
(w), K is a finite rank operator that commutes with T . Then T + K satisfies
property (w).
A similar result holds for nilpotent perturbations.
Theorem 4.4. [3] Suppose that T ∈ L(X) is a-isoloid. If T satisfies property
(w) and N is nilpotent operator that commutes with T then T + N satisfies
property (w).
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 55–72
Weyl’s type theorems and perturbations
69
Example 4.5. Both Theorem 4.4 and Theorem 4.3 fail if we assume that
the nilpotent operator N , and the finite rank operator K do not commute
with T . For instance, let T and N be as in Example 2.5. Then T satisfies
Weyl’s theorem and is decomposable, since T is a quasi-nilpotent operator.
This implies that T satisfies property (w), by Corollary 2.10 of [10].
On the other hand, T + N does not satisfy Weyl’s theorem, and consequently, by the implication (12), T + N does not satisfy property (w).
Example 4.6. The following example shows that Theorem 4.3 fails if we do
not assume that T is a-isoloid. Let T and K be defined as in Example 2.6.
Clearly, α(T ) = 0 so T is not a-isoloid. Now, T satisfies Weyl’s theorem, and
hence by Theorem 2.16 of [10] T satisfies property (w).
On the other hand, Weyl’s theorem does not hold for T + K and this
implies (see (12)) that property (w) fails for T + K.
Example 4.7. In general, property (w) is not transmitted from T to a quasinilpotent perturbation T + Q. For instance, take T = 0, and Q ∈ L(ℓ2 (N))
defined by
x2 x3
Q(x1 , x2 , . . . ) = ( , , . . . ) for all (xn ) ∈ ℓ2 (N,
2 3
Then Q is quasi-nilpotent and {0} = π00 (Q) 6= σa (Q) \ σaw (Q) = ∅ . Hence
T satisfies property (w) but T + Q = Q fails this property.
However, property (w) is preserved under injective quasi-nilpotent perturbations.
Theorem 4.8. [3] Suppose that T ∈ L(X) and Q is an injective quasinilpotent operator commuting with T . If T satisfies property (w) then also
T + Q satisfies property (w).
Property (w) is preserved under finite rank perturbations or nilpotent
perturbations commuting with T in the case that T ∗ ∈ L(X) has property
H(p).
Theorem 4.9. [3] Suppose that T ∗ ∈ L(X) has property H(p), K is either a
finite rank operator or a nilpotent operator commuting with T . Then property
(w) holds for T + K.
An analogous result holds for paranormal operators acting on Hilbert
spaces.
Theorem 4.10. [3] Suppose that T ′ ∈ L(H) is paranormal and K is either a
finite rank operator or a nilpotent operator commuting with T . Then property
(w) holds for T + K.
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Pietro Aiena, Jesús R. Guillen, Pedro Peña
Recall that every generalized scalar operator satisfies property (w) [10,
Corollary 2.11].
Theorem 4.11. [3] Suppose that T ∈ L(X) is generalized scalar operator, K
an algebraic operator commuting with T . Then property (w) holds for T + K.
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